The spectral density matrix is a fundamental object of interest in time series analysis, and it encodes both contemporary and dynamic linear relationships between component processes of the multivariate system. In this paper we develop novel inference procedures for the spectral density matrix in the high-dimensional setting. Specifically, we introduce a new global testing procedure to test the nullity of the cross-spectral density for a given set of frequencies and across pairs of component indices. For the first time, both Gaussian approximation and parametric bootstrap methodologies are employed to conduct inference for a high-dimensional parameter formulated in the frequency domain, and new technical tools are developed to provide asymptotic guarantees of the size accuracy and power for global testing. We further propose a multiple testing procedure for simultaneously testing the nullity of the cross-spectral density at a given set of frequencies. The method is shown to control the false discovery rate. Both numerical simulations and a real data illustration demonstrate the usefulness of the proposed testing methods.

翻译：谱密度矩阵是时间序列分析中的一个基本研究对象，它编码了多元系统中各分量过程之间的同期和动态线性关系。本文针对高维情形下的谱密度矩阵发展了新的推断方法。具体而言，我们引入了一种新的全局检验程序，用于检验给定频率集合上跨分量指标的交叉谱密度是否为零。首次将高斯近似和参数自助法两种方法论应用于频率域中高维参数的推断，并开发了新的技术工具，以提供全局检验在水平精度和功效上的渐近保证。我们进一步提出了一种多重检验程序，用于同时检验给定频率集合上交叉谱密度的零值性。该方法被证明能够控制错误发现率。数值模拟和实际数据实例均展示了所提出的检验方法的实用性。